Today we’re going to begin a dive into trigonometry. Eventually we’ll talk about everything from right triangle trig to the unit circle to trig functions. For now, let’s start at the beginning: Triangle and Trigonometry Basics.

## Triangle Basics:

In order to talk about trigonometry, we have to be on the same page about certain facts and standard ways of thinking about triangles. Here are some of them:

### Angles

The three angles in a triangle add to 180°. A **right angle** is a 90° angle, and a **right triangle** is a triangle containing a right angle. In a right triangle, the two angles that are not the right angle must add to 90°. Those angles are both called **acute**, meaning that their measure is less than 90°.

### Naming Conventions

For our purposes, we’ll represent the angles of a triangle with capital letters, usually A, B, and C, and the sides with lower case letters, usually a, b, and c. Angle A is formed by two of the sides of the triangle; side a is “**opposite**” that angle, or the one side that doesn’t touch angle A. Same with the other sides and angles – if they share a letter, they are opposite each other.

*Neat and useful fact*: The smallest angle in a triangle is opposite the shortest side. The largest angle is opposite the longest side. And, if you like, the “middlest” angle is opposite the side of middle length.

### Finding Right Triangles

There are a few ways to know that a triangle is a right triangle. First, if you know that a specific angle has a measure of 90°, it’s a right triangle! Sometimes you’ll see a little square drawn into the triangle, as you see at angle C on the right. That little square means that the two lines it touches are perpendicular, another way of telling you that the angle with the little square is a right angle.

### Opposite, Adjacent, Hypotenuse

In right triangles, we can use opposite, adjacent, and hypotenuse to talk about the sides of the triangle *relative* to a particular angle. The names of the sides change if we focus on a different angle. We only use these words when focusing on the acute angles; never the 90° angle.

You already know what I mean if I refer to the side **opposite** of angle B – it’s the side of a triangle that doesn’t touch angle B. The **hypotenuse** is the side opposite the right angle. The hypotenuse is the longest side in a right triangle because it’s opposite the biggest angle. The third side, which we haven’t named opposite or hypotenuse, is **adjacent** to angle B. The adjacent side is the side touching the angle that is not the hypotenuse. In the diagram above, side b is opposite angle B, side c is the hypotenuse, and side a is adjacent to angle B. If we focus on angle A instead of angle B: side a is opposite angle A, aide c is the hypotenuse, and side b is adjacent to angle A.